satisfying \(\beta \) -geo-indistinguishability, as proposed by Ghosh et al. in [1]}. Flipping the roles of \(x\)  and \(y\) , considering \(\pi _{\mathcal {X}}\)  to be the empirical distribution of observed locations obfuscated by \(\mathcal {C}_{\text{geo}}\) , we reduce (REF ) to the iterative step of IBU. As we consider a finite space of locations, we advantage from the fact that BA converges and, therefore, implying that \(\lim \limits _{t\rightarrow \infty }c_t\)  exists and, hence, is unique. This equivalence with IBU's iterative step shows that \(\lim \limits _{t\rightarrow \infty }c_t\)  gives a unique MLE for \(\pi _{\mathcal {X}}\)  under \(\mathcal {C}_{\text{geo}}\) , i.e., \(\lim \limits _{t\rightarrow \infty }c_t=\hat{\pi }_{\mathcal {X}}\) . Thus, we discover a wonderful duality between IBU and BA, and exploit this to obtain a new approach to prove following theorem, complementing the proof given by ElSalamouny et al. for Corollary 2 in [2]}.
